Sampling and Complexity of Partition Function

Abstract: The number partition problem is a well-known problem, which is one of 21 Karp's NP-complete problems \cite{karp}. The partition function is a boolean function that is equivalent to the number partition problem with number range restricted. To fully understand the computational complexity of the number partition problem and the partition function is quite important and hard. People speculate that we need new tools and methods \cite{aaronson} for such problem. In our recent research on the universal learning machine \cite{paper5, paper8}, we developed some tools, namely, fitting extremum, proper sampling set, boolean function with parameters (used in trial-and-error fashion). We found that these tools could be applied to the partition function. In this article, we discuss the set up of the partition function, properties of the partition function, and the tools to be used. This approach leads us to prove that the lower bound of the computational complexity of partition function, as well as the lower bound of the computational complexity of the number partition problem, is exponential to the size of problem. This implies: P $\ne$ NP \cite{cook}.